Note N. 138 — SECTION VIII.

Note 138, p. 60. Suppose that it were required to find the position of a point in space, as of a planet, and that one observation places it in n, fig. 34, another observation places it in nʹ, another in nʺ, and so on; all the points n, nʹ, nʺ, nʹʹʹ, &c., being very near to one another. The true place of the planet P will not differ much from any of these positions. It is evident, from this view of the subject, that P n, P nʹ, P nʺ, &c., are the errors of observation. The true position of the planet P is found by this property, that the squares of the numbers representing the lines P n, P nʹ, &c., when added together, is the least possible. Each line P n, P nʹ, &c., being the whole error in the place of the planet, is made up of the errors of all the elements; and, when compared with the errors obtained from theory, it affords the means of finding each. The principle of least squares is of very general application; its demonstration cannot find a place here; but the reader is referred to Biot’s Astronomy, vol. ii. p. 203.